8657pre.qxd 05/07/2006 08:24 Page i Children’s Mathematics 8657pre.qxd 05/07/2006 08:24 Page ii 8657pre.qxd 05/07/2006 08:24 Page iii Children’s Mathematics Making Marks, Making Meaning Second Edition Elizabeth Carruthers and Maulfry Worthington 8657pre.qxd 05/07/2006 08:24 Page iv ᭧ Elizabeth Carruthers and Maulfry Worthington 2006 First published 2006 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act, 1988, this publication may be reproduced, stored or transmitted, in any form or by any means, only with the prior permission in writing of the publishers, or in the case of reprographic reproduction, in accordance with the terms of licences issued by the Copyright Licensing Agency. Inquiries concerning reproduction outside those terms should be sent to the publishers. Paul Chapman Publishing A SAGE Publications Company 1 Oliver’s Yard London EC1Y 1SP SAGE Publications Inc 2455 Teller Road Thousand Oaks, California 91320 SAGE Publications India Pvt Ltd B-42, Panchsheel Enclave Post Box 4109 New Delhi 110 017 Library of Congress Control Number: 2006923703 A catalogue record for this book is available from the British Library ISBN 10 1-4129-2282-8 ISBN 13 978-1-4129-2282-1 ISBN 10 1-4129-2283-6 ISBN 13 978-1-4129-2283-8 (pbk) Typeset by Dorwyn, Wells, Somerset Printed in Great Britain by T.J. International, Padstow, Cornwall Printed on paper from sustainable resources 8657pre.qxd 05/07/2006 08:24 Page v Contents About the Authors ix Acknowledgements xi Foreword by John Matthews xiii Foreword by Chris Athey xv Preface xvii 1 Who takes notice of children’s own ‘written’ mathematics? 1 • Children’s mathematical graphics 2 • International findings 3 • Studies that relate to mathematical literacy 9 • Enquiring into children’s mathematics 11 2 Making marks, making meaning 13 • Children making meaning with marks 13 • Different literacies: mathematical literacy 14 • Children represent their mathematical actions and understanding on paper 14 • Learning theories 20 • Reading and using mathematical graphics 25 • Socio-cultural contexts in Early Years settings 31 • Teachers’ beliefs 32 • Creativity in mathematics 34 • Summary 34 3 Mathematical schemas 36 • What is a schema? 36 • Schemas and mathematics 40 • Schemas and mark-making 41 • Observing schemas in a school setting 44 • Mapping patterns of schema exploration 51 v 8657pre.qxd 05/07/2006 08:24 Page vi vi Children’s Mathematics 4 Early writing, early mathematics 56 • The significance of emergent writing 57 • Young children explore symbols 58 • Early writing and early mathematical marks 63 • Early (emergent) literacy is often misunderstood 66 • Conclusion 68 5 Bridging the gap between home and school mathematics 69 • Disconnections 69 • Understanding symbols 72 • Mathematics as a foreign language 77 • Becoming bi-numerate 79 • Teachers’ difficulties 82 • Conclusion 83 6 Making sense of children’s mathematical graphics 84 • The evolution of children’s early marks 84 • Categories of children’s mathematical graphics 86 • Common forms of graphical marks 87 • Early development of mathematical meaning 91 • Early explorations with marks 93 • ‘The beginning is everything’ 95 • Early written numerals 96 • Numerals as labels 99 • Representations of quantities and counting 100 • The development of early written number, quantities and counting 105 7 Understanding children’s developing calculations 106 • Practical mathematics 106 • The fifth dimension: written calculations 108 • Representations of early operations 108 • Counting continuously 109 • Narrative actions 112 • Supporting children’s own mathematical marks 114 • Separating sets 117 • Exploring symbols 118 • Standard symbolic calculations with small numbers 123 • Calculations with larger numbers supported by jottings 126 • The development of children’s mathematical graphics: becoming bi-numerate 130 • Conclusion 132 8657pre.qxd 05/07/2006 08:24 Page vii Contents vii 8 Environments that support children’s mathematical graphics 134 • Rich mathematical environments for learning 134 • The balance between adult-led and child-initiated learning 136 • Role-play and mark-making 139 • The physical environment 140 • Practical steps 145 • Graphics areas 149 • Conclusion 161 9 Case studies from early childhood settings 162 • The birthday cards 162 • A number line 164 • ‘No entry’ 166 • Carl’s garage 167 • Children’s Centres: The Cambridge Learning Network project 169 • The spontaneous dice game 172 • Young children think division 174 • A zoo visit 177 • Mathematics and literacy in role-play: the library van 178 • Aaron and the train 181 • Multiplying larger numbers 185 • Nectarines for a picnic 187 • Conclusion 190 10 Developing children’s written methods 192 • The assessment of children’s mathematical representations on paper 192 • The problem with worksheets 194 • Assessing samples of children’s own mathematics 197 • Examples of assessment of children’s mathematics 199 • The pedagogy of children’s mathematical graphics 204 • Modelling mathematics 205 11 Involving parents and families 216 • Children’s first and continuing educators 216 • The home as a rich learning environment 217 • What mathematics do young children do at home? 218 • What mathematics do parents notice at home? 221 • Parents observe a wealth of mathematics 225 • Helping parents recognise children’s mathematical marks 225 • Parents’ questions about children’s mathematical graphics 226 • Conclusion 227 8657pre.qxd 05/07/2006 08:24 Page viii viii Children’s Mathematics 12 Children, teachers and possibilities 229 • Inclusion 229 • Children’s questions 230 • Teachers’ questions 231 • It’s all very well – but what about test scores? 234 Reflections 236 Appendix: our research 238 Glossary 240 References 243 Author Index 253 Subject Index 256 8657pre.qxd 05/07/2006 08:24 Page ix About the Authors Elizabeth Carruthers and Maulfry Worthington have each taught in the full 3–8 year age range for over 25 years. Early in their careers both developed incurable cases of curiosity and enthusiasm in Early Years education which fails to diminish. They have carried out extensive research in key aspects of Early Years education, with a partic- ular focus on the development of children’s mathematical graphics from birth – eight years. Publications include articles, papers and chapters on the development of mathematical understanding. Elizabeth Carruthers is presently head teacher of the Redcliffe Integrated Children’s Centre in Bristol. She has recently worked within an Early Years Advisory Service in a local authority and as a National Numeracy Consultant. Elizabeth has been a mentor with the Effective Early Learning Project (EEL) and has lectured on Early Childhood courses. She has taught and studied in the United States and is currently working on her doctorate researching mathematical graphics and pedagogical approaches. Elizabeth is an advocate for the rights of teenage cancer patients and a supporter of the Teenage Cancer Trust. Maulfry Worthington is engaged in research for her doctorate on multi-modality within children’s mathematical graphics (Free University, Amsterdam): she also works as an independent Early Years consultant. Maulfry has worked as a National Numeracy Consultant and has lectured in Initial Teacher Education on Primary and Early Years mathematics, Early Years pedagogy and Early Years literacy. She has also worked at the National College for School Leaders as an e-learning facilitator on a number of Early Years online communities and programmes. Maulfry and Elizabeth are Founders of the international Children’s Mathematics Network, established in 2003, described on their website as: ‘an international, non-profit-making organization for teachers, practitioners, stu- dents, researchers and teacher educators working with children in the birth–8 year age range. It is a grassroots network, with children and teachers at the heart of it and focuses on children’s mathematical graphics and the meanings children make. ix 8657pre.qxd 05/07/2006 08:24 Page x x Children’s Mathematics Early ‘written’ mathematics is explored within the context of visual representation including drawing; early (emergent) writing; schemas; play; thinking; creativity and multi-modal meanings. Our work is based on extensive, evidence-based research with children, teachers and families and within the context of homes, nurseries and schools. We advocate a spirit of freedom and creativity for teachers and more importantly, the freedom for children to explore their own meanings in creative ways. Our aim is to hear the voice of the child. (See the website at www.childrens-mathematics.net.) Elizabeth and Maulfry are winners of several national awards for their work on math- ematical graphics with children and with teachers including TACTYC’s 2003 Jenefer Joseph Award for the ‘Creative Arts in the Early Years’ (3–8), and were shortlisted for Becta’s ICT in Practice Award in the ‘Innovation and Change’ category, 2004. Dedication We dedicate this book to our own creative children: Mhairi, Sovay, Laura and Louise, and to the memory of two strong women – our mothers, Elizabeth Gillon Carruthers and Muriel Marianne Worthington. 8657pre.qxd 05/07/2006 08:24 Page xi Acknowledgements We should like to pay tribute to all the adults and children who contributed to our thinking about children’s mathematics. Our sincere thanks go in particular to Chris Athey who, through her writing, really helped us observe and understand young